Srinivas Bounded Mathematics: A Formal Axiomatic Framework

This paper introduces Srinivas Bounded Mathematics (SBM), a formal axiomatic framework in which all mathematical constructions, operations, and processes are explicitly bounded and finite. The system is developed through eight rigorous axioms that prohibit infinite subdivision while establishing context-dependent finite lower bounds (AR Bounds). A central innovation is the distinction between traditional zero (0, the unreachable mathematical ideal) and SBM Absolute Zero (◦, the absorbing state for sub-threshold positive quantities), where ◦ > 0. This recognition transforms classical problems: ◦ ÷ ◦ = 1 follows directly, while traditional 0 ÷ 0 is established as ill-posed within SBM—an independence result. This version (v3.0) is prepared for journal submission and includes: (1) the Non-Zero Measurability Theorem with complete formal derivation from axioms, establishing that all measurable quantities must satisfy |Q| ≥ ε_AR > 0 as a structural result independent of any specific numerical value; (2) philosophical discussion on formalism versus human-guided mathematical structure; (3) the Definability Boundary concept explaining that quantities below AR-absolute are ill-defined non-entities; (4) the symmetry between large numbers and small numbers (LNGN ↔ 10^(-LNGN)); (5) explanation of why quantization is forced by the nature of definability itself; (6) detailed analysis of Axiom 8 as the dynamic quantization engine; (7) explicit extensibility of AR Bounds through custom context-specific bounds. The framework resolves classical paradoxes (Gabriel's Horn, Zeno's Dichotomy, Banach-Tarski) as natural consequences of bounded subdivision, establishes finite-step convergence of Cauchy sequences, and maintains compatibility with constructive mathematics while providing natural correspondence with physical bounds observed in empirical sciences. SBM is presented as a legitimate alternative mathematical foundation, analogous to how non-Euclidean geometry emerged from modifying Euclid's parallel postulate.